Quantised cavity modes

We consider a planar optical cavity that is perfectly reflecting, with the cavity walls perpendicular to the z axis [133] as shown in Figure 5.1. In this case, the magnetic vector potential can be expanded over a set of orthonormal modes with polarisation η ∈ {1, 2} and wavevector

κl_k= (kx, ky, πl/L). (5.2)

The wavenumbers kx and ky are good quantum numbers as they lie in the transitionally

invariant plane of the cavity, whereas the z-component is quantised with l ∈ N. We pa- rameterise κ in spherical coordinates as θl

k= arccos πl/L|κlk|
and φk= arctan (ky/kx),

and denote the wavevector in the plane of the cavity (which is also the plane of the array) as k = (kx, ky). We are only interested in the value of the vector potential in the plane of

the cavity where we shall embed the honeycomb array, which we choose to be the centre of the cavity at z = L/2 as shown in Figure 5.1. Thus, we define A(ρ) ≡ A(ρ, L/2) where ρ = (x, y).

We find that even multiples of l correspond to modes polarised in the ˆz direction, whereas odd multiples of l correspond to modes polarised in the plane of the array. We separate A into these two parts

A = A⊥+ Ak, (5.3) where A⊥(ρ) = −ω0ˆz even X l Nl X k sin θ_klω_kl−1/2c_kleik·ρ+ H.c., (5.4)

Chapter 5 - Chirality inversion in Dirac polaritons 98

Figure 5.1: Schematic of (a section of) the cavity and embedded nanoparticle array. The top-right inset displays an alternative viewing angle.

Ak(ρ) = odd X l Nl−1 X k,η uklηω −1/2 kl dklηeik·ρ+ H.c. (5.5)

The operators c†_kl and d†_klη create cavity photons with dispersion ωkl= c|κlk|, polarised

along the direction ˆz and uklη respectively, where

uklη =     

i cos θ_kl(cos φkˆx + sin φkˆy), η = 1

− sin φ_kˆx + cos φkˆy, η = 2

. (5.6)

Here V is the volume of the cavity and Nl is a normalisation constant given by

Nl=      (~/20V)1/2, l = 0 (~/0V)1/2(−1)l/2, l 6= 0 . (5.7)

Note that for odd l we sum over the two polarisations in the vector potential of equation

5.3, whereas for even l only the TM polarisation has a non-vanishing field in the centre of the cavity where we embed the array. In the following section we will consider dipoles polarised in the ˆz direction, originating from nanoparticles orientated perpendicular to the cavity array with an anisotropy sufficient to decouple the transverse and longitudinal

Chapter 5 - Chirality inversion in Dirac polaritons 99

Figure 5.2: Plots showing the dispersion of the TEM mode l = 0 (red line) and the TM cavity modes l 6= 0 (black curves) for various cavity heights as indicated in the figures. We see that as the cavity height is decreased the dispersions of the TM modes are shifted up in frequency, whilst the dispersion of the TEM mode is independent of the cavity height. In the figure c/ω0a = 3/5, D= 1 and a = 3r.

modes within the frequency bandwidth of interest. In this case we need just consider A⊥.

There are some distinguishing features between the TEM mode l = 0 and the other TM cavity modes l 6= 0. As we see in Figure 5.2 the dispersion of the TEM mode is independent of the cavity height, whereas the TM cavity modes shift up in frequency with decreasing L. Therefore, the frequency of the TEM mode ωk = c|k| is always less

than the frequency of a TM cavity mode ωkl. In addition the light-matter coupling

between the CPs and the TM cavity modes vanishes as L → 0. Therefore, the existence of the TEM mode is crucial as it permits tuning of the light-matter interaction without the detuning that occurs for the TM cavity modes, and maintains a non-vanishing light- matter interaction as the cavity height is decreased. In fact to capture the essential properties of the polaritons we need only consider the TEM mode, as the inclusion of the higher order l 6= 0 modes provides only a qualitative modification of the band structure and eigenstates. For this reason we consider just the fundamental l = 0 (TEM) cavity mode to highlight the key physics. Under this assumption the magnetic vector potential reads A(ρ) = −ˆzX k  ~ 20Vωk 1/2 c_keik·ρ+ H.c. (5.8)

Furthermore in anticipation of the following section we can express the wavevector as

Chapter 5 - Chirality inversion in Dirac polaritons 100

where q is restricted to the first Brillouin zone of the honeycomb array, and Gn is an

arbitrary reciprocal lattice vector labelled by n. Thus, we write equation 5.8as

A(ρ) = −ˆzX qn  ~ 20Vωqn 1/2 c_qnei(q−Gn)·ρ_{+ H.c., (5.10)}

where ωqn= c|q − Gn| and c†qn creates a cavity photon with wavevector q − Gn. Note

that in equation 5.10 we have not made any physical changes to A with respect to equation 5.8, and in the context of an empty cavity we can regard our transformation as just a relabelling of k.